Editing Hilbert's fourth problem (section)

==Generalizations of Hilbert's fourth problem==
There was found the correspondence  between the planar ''n''-dimensional Finsler metrics and special symplectic forms on the Grassmann manifold <math>G(n+1,2)</math> в <math>E^{n+1}</math>.<ref>{{cite journal
| last1=Álvarez Paiva | first1=J. C.
| title=Sympletic geometry and Hilbert fourth problem
| journal=Journal of Differential Geometry
| volume=69
| issue=2
| date=2005
| pages=353–378
| doi=10.4310/jdg/1121449109 | doi-access=free}}</ref>

There were considered periodic solutions of  Hilbert's fourth problem :
# Let (''M'', ''g'') be a compact locally Euclidean Riemannian manifold. Suppose that <math>C^2</math> Finsler metric on ''M'' with the same  geodesics  as in the metric ''g'' is given. Then  the Finsler metric is the sum of  a locally Minkovski metric and a closed 1-form.<ref name="Paiva2018">{{cite journal
| last1=Álvarez Paiva | first1=J. C.
| last2=Barbosa Gomes | first2=J.
| title=Periodic Solutions of Hilbert fourth problem
| journal=Journal of Topology and Analysis
| arxiv=1809.02783
| date=2018
| doi=10.1142/S1793525321500576| s2cid=240026741
}}</ref>
# Let (''M'', ''g'') be a compact symmetric Riemannian space of rank greater than one. If ''F'' is a symmetric <math>C^2</math> Finsler metric whose geodesics coincide with  geodesics of the Riemannian metric ''g'', then (''M'', ''g'') is a symmetric Finsler space.<ref name="Paiva2018"/> The analogue of this theorem for rank-one symmetric spaces has not been proven yet.

Another exposition of Hilbert's fourth problem can be found in work of Paiva.<ref>{{cite journal
| last1=Álvarez Paiva | first1=J. C.
| title=Hilbert's fourth problem in two dimensions
| journal=MASS Selecta
| date=2003
| pages=165–183}}</ref>